Abstract
Using the backprojection filtration (BPF) and filtered backprojection (FBP) approaches, respectively, we prove that with cone-beam CT the interior problem can be exactly solved by analytic continuation. The prior knowledge we assume is that a volume of interest (VOI) in an object to be reconstructed is known in a subregion of the VOI. Our derivations are based on the so-called generalized PI-segment (chord). The available projection onto convex set (POCS) algorithm and singular value decomposition (SVD) method can be applied to perform the exact interior reconstruction. These results have many implications in the CT field and can be extended to other tomographic modalities, such as SPECT/PET, MRI.
Highlights
In 2002, an exact and efficient helical cone-beam reconstruction method was developed by Katsevich [1, 2], which is a significant breakthrough in the area of helical/spiral conebeam CT
The Katsevich formula is in a filtered backprojection (FBP) format using data from a PI-arc corresponding to the so-called PI-segment
Inspired by the tremendous biomedical implications including local cardiac CT at minimum dose, local dental CT with high accuracy, CT guided procedures, nano-CT, and so on [18], we recently proved, using analytic continuation, that the interior problem can be exactly solved if a subregion in an region of interest (ROI) in the field of view (FOV) is known [19, 20], while the conventional wisdom is that the interior problem does not have a unique solution [21]
Summary
In 2002, an exact and efficient helical cone-beam reconstruction method was developed by Katsevich [1, 2], which is a significant breakthrough in the area of helical/spiral conebeam CT. For important biomedical applications including bolus-chasing CT angiography [4] and electronbeam CT/micro-CT [5], our group obtained the first proofs of the general validities of both the BPF and FBP formulae in the case of cone-beam scanning along a general smooth scanning trajectory [6,7,8,9]. A recent milestone is the two-step Hilbert transform method developed by Noo et al [16] In their framework, an object image on a PI-line/chord can be exactly reconstructed if the intersection between the chord and the object is completely covered by a field of view (FOV). We will present further ideas and conclude the paper
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